Hey there! We receieved your request
Stay Tuned as we are going to contact you within 1 Hour
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-5470-145
+91 7353221155
Use Coupon: CART20 and get 20% off on all online Study Material
Complete Your Registration (Step 2 of 2 )
Sit and relax as our customer representative will contact you within 1 business day
OTP to be sent to Change
Another useful combination of two functions f and g is the composition of these two functions. Let f : X → Y and g : Y → Z be two functions.
We define a function h : X → Z by setting h(x) = g(f(x). To obtain h(x), we first take the f-image f(x), of an element x in X so that f(x) ε Y, which is the domain of g(x) and then take the g-image of f(x), that is, g(f(x)), which is an element of Z. The scheme is shown in the figure.
The function h, defined above, is called the composition of f and g and is written gof. Thus (gof)(x) = g(f(x)). Domain of gof = {x : x in domain f, f(x) in domain g}.
e.g. Let f : R → R be a function defined by f(x) = x2 + 4 and g[0, ∞) → R be a function defined by g(x) = √x. Then gof(x) = g(f(x)) = √(x2 + 4). Domain of gof = R. Thus we have gof : R → R defined by (gof)(x) = √(x2 + 4). Similarly, we shall have fog : [0, ∞) → R defined by (fog)(x) = x + 4. Note that (gof)(x) ≠ (fog)(x).
Illustration: Two functions are defined as under:
Find fog and gof.
Solution: (fog)(x) = f(g(x))
Let us consider, g(x) < 1 :
(i) x2 < 1, -1 < x < 2 => -1 < x < 1, -1 < x < 2 => -1 < x < 1
(ii) x2 + 2 < 1, 2 < x < 3 => x < -1, 2 < x < 3 => x = φ
Let us consider, 1 < g(x) < 2,
(iii) 1 < x2 < 2, -1 < x < 2
=> x ε [-√2, -1) υ (1,√2] , -1 < x < 2 => 1 < x < √2
(iv) 1 < x+2 < 2, 2 < x < 3 => -1 < x < 0, 2 < x < 3, x = φ
Let us consider -1 < f(x) < 2 :
(i) -1 < x+1 < 2, x < 1 => -2 < x < 1, x < 1 => -2 < x < 1
(ii) -1 < 2x+1 < 2, 1 < x < 2 => -1 , x < ½, 1 < x < 2 => x= φ
Let us consider 2 < f(x) < 3:
(iii) 2 < x+1 < 3 , x < 1 => x < 2 , x < 1 => x = 1
(iv) 2 < 2x+1 < 3, 1 < x < 2 => 1 < 2x < 2, 1 < x < 2
=> ½ < x < 1 , 1 < x < 2 => x = φ
If we like we can also write g(f(x)) = (x+1)2, -2 < x < 1.
To read more, Buy study materials of Set Relations and Functions comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
Get your questions answered by the expert for free
You will get reply from our expert in sometime.
We will notify you when Our expert answers your question. To View your Question
Composite Functions Problem of finding out fog and...
Bounded and Unbounded Function Let a function be...
Set Theory Table of Content Set Union of Sets...
Graphical Representation of a Function The...
Greatest Integer Function The function f(x) : R...
Graphical Representation of a Function 10....
Constant Function and the Identity Function The...
Absolute Value Function The function defined as:...
Basic Transformations on Graphs Drawing the graph...
Exponential Function Exponential and Logarithmic...
Signum Function The signum function is defined as...
Explicit and Implicit Functions If, in a function...
Periodic Function These are the function, whose...
Even and Odd Function A function f(x) : X → Y...
Logarithmic Function We have observed that y = a x...
Invertible Function Let us define a function y =...
Functions Table of Content What are Functions?...
Increasing or Decreasing Function The function f...
Introduction to Functions Definition of Function:...
Inverse Function Let f : X → Y be a function...
Set, Relations and Functions – Solved...
Linear Function When the degree of P(x) and Q(x)...
Polynomial and Rational Function A function of the...
Relations Table of Content What do we mean by...
Cartesian Product of Sets Table of Content Define...
Algebra of Functions Given functions f : D →...
Functions: One-One/Many-One/Into/Onto Functions...